Section 1.6. Quotient Groups 47 (iii) Prove that −I is the only element in Q of order 2, and that all other elements M = I satisfy M 2 = −I. Conclude that Q has a unique subgroup of order 2, namely, −I , and it is the center of Q. 1.68. Prove that the elements of Q can be relabeled as ±1, ±i, ±j, ±k, where i2 = j2 = k2 = −1, ij = k, jk = i, ki = j, ij = −ji, ik = −ki, jk = −kj. 1.69. Prove that the quaternions Q and the dihedral group D8 are nonisomorphic groups of order 8. 1.70. Prove that A4 is the only subgroup of S4 of order 12. 1.71. (i) For every group G, show that the function Γ: G Aut(G), given by g γg (where γx is conjugation by g), is a homomorphism. (ii) Prove that ker Γ = Z(G) and im Γ = Inn(G) conclude that Inn(G) is a subgroup of Aut(G). (iii) Prove that Inn(G) Aut(G). Section 1.6. Quotient Groups The construction of the additive group of integers modulo m is the prototype of a more general way of building new groups, called quotient groups, from given groups. The homomorphism π : Z Im, defined by π : a [a], is surjective, so that Im is equal to im π. Thus, every element of Im has the form π(a) for some a Z, and π(a) + π(b) = π(a + b). This description of the additive group Im in terms of the additive group Z can be generalized to arbitrary, not necessarily abelian, groups. Suppose that f : G H is a surjective homomorphism between groups G and H. Since f is surjective, each element of H has the form f(a) for some a G, and the operation in H is given by f(a)f(b) = f(ab), where a, b G. Now ker f is a normal subgroup of G, and the First Isomorphism Theorem will reconstruct H = im f and the surjective homomorphism f from G and ker f alone. We begin by introducing a binary operation on the set S(G) of all nonempty subsets of a group G. If X, Y S(G), define XY = {xy : x X and y Y }. This multiplication is associative: X(Y Z) is the set of all x(yz), where x X, y Y , and z Z, (XY )Z is the set of all such (xy)z, and these are the same because (xy)z = x(yz) for all x, y, z G. Thus, S(G) is a semigroup in fact, S(G) is a monoid, for {1}Y = {1 · y : y Y } = Y = Y {1}. An instance of this multiplication is the product of a one-point subset {a} and a subgroup K G, which is the coset aK. As a second example, we show that if H is any subgroup of G, then HH = H.
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